On the Hausdorff dimension of certain sets arising in Number Theory

Details On the Hausdorff dimension of certain sets arising in Number Theory On the Hausdorff dimension of certain sets arising in Number Theory

1999-08-10

arxiv.org/abs/math/9908043v3

Mathematics

Number Theory

We add a reference and we correct the Hausdorff dimension of E_2

Scientific paper

This paper has been withdrawn Any real number $x$ in the unit interval can be expressed as a continued fraction $x=[n_1,...,n_{_N},...]$. Subsets of zero measure are obtained by imposing simple conditions on the $n_{_N}$. By imposing $n_{_N}\le m \forall N\in \zN$, Jarnik defined the corresponding sets $E_m$ and gave a first estimate of $d_H(E_m)$, $d_H$ the Hausdorff dimension. Subsequent authors improved these estimates. In this paper we deal with $d_H(E_m)$ and $d_H(F_m)$, $F_m$ being the set of real numbers for which ${\sum_{i=1}^N n_i\over N}\le m$.

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